Problem Statement
In levelling across a river, two pegs A and B were fixed on opposite banks. The following readings were taken:
| Instrument Position | Staff Reading (m) | |
|---|---|---|
| At Point A | At Point B | |
| Level at A | 1.871 | 1.469 |
| Level at B | 1.664 | 0.706 |
If R.L. of A is 50.865, find the R.L. of the point B.
Step-by-Step Solution
Key Information & Setup
- Staff reading at A (instrument at A) = 1.871 m
- Staff reading at B (instrument at A) = 1.469 m
- Staff reading at A (instrument at B) = 1.664 m
- Staff reading at B (instrument at B) = 0.706 m
- R.L. of point A = 50.865 m
- Goal: Find the R.L. of point B using reciprocal leveling
Step 1: Calculate Apparent Difference in Elevation with Instrument at A
The apparent difference in elevation between points can be calculated as:
Apparent difference in elevation = Staff reading at A – Staff reading at B
Apparent difference in elevation = 1.871 – 1.469 = 0.402 m
Since the staff reading at A is greater than at B, point B appears higher than point A in this observation.
Apparent difference in elevation (instrument at A) = 0.402 m (B being higher)
Step 2: Calculate Apparent Difference in Elevation with Instrument at B
Similarly, we calculate the apparent difference when the instrument is at point B:
Apparent difference in elevation = Staff reading at A – Staff reading at B
Apparent difference in elevation = 1.664 – 0.706 = 0.958 m
Again, since the staff reading at A is greater than at B, point B appears higher than point A in this observation as well.
Apparent difference in elevation (instrument at B) = 0.958 m (B being higher)
Step 3: Calculate True Difference in Elevation between Points
In reciprocal leveling, the true difference in elevation is the mean of the apparent differences:
True difference in elevation = (Apparent difference 1 + Apparent difference 2) ÷ 2
True difference in elevation = (0.402 + 0.958) ÷ 2
True difference in elevation = 1.360 ÷ 2 = 0.680 m
Since both apparent differences indicate that B is higher than A, the true difference confirms that point B is higher than point A by 0.680 m.
True difference in elevation = 0.680 m (B being higher than A)
Step 4: Calculate the R.L. of Point B
Using the known R.L. of point A and the true difference in elevation:
R.L. of point B = R.L. of point A + True difference in elevation
R.L. of point B = 50.865 + 0.680
R.L. of point B = 51.545 m
Final Result
Explanation of River Crossing Leveling
Importance of Reciprocal Leveling for River Crossings:
- Overcoming Obstacles: Rivers present natural barriers that make conventional leveling impossible or impractical.
- Error Compensation: Reciprocal leveling minimizes errors caused by refraction and earth’s curvature, which are particularly significant over water bodies.
- Higher Precision: By averaging readings from both sides, random errors in observation are reduced.
- Practical Application: Essential for designing bridges, culverts, pipelines, and other structures that cross water bodies.
Formula for Reciprocal Leveling:
True difference in elevation = (h₁ + h₂) ÷ 2
Where h₁ and h₂ are the apparent differences from each instrument position
The reciprocal leveling method is particularly valuable in surveying projects that require precise elevation measurements across rivers, deep valleys, or other terrain features where direct leveling would introduce substantial errors or be physically impossible.
